The Hardy inequalities (Q1316929)

Using the Fourier transform, the author proves that \[ \| f^{(k)} \| \leq {n - k \over k} h^{ - k} \cdot 2^{ - m} \bigl \| \Delta^ m_{\pi h} (f) \bigr \| + {k \over n} h^{n - k} \| f^{(n)} \| \] for any \(f \in L_ 2 (\mathbb{R})\) with an absolutely continuous derivative \(f^{(n - 1)}\). Here \(0 < m < k < n\) are given integers, \(\| \cdot \|\) is the norm in \(L_ 2 (\mathbb{R})\) and the symbol \(\Delta^ m_{\pi h}\) \((h > 0)\) stands for the difference operator of the order \(m\) with the step \(\pi h\).